Problem 15 IYPT 2027: 15. Cold drink

Problem Description & Analysis

A can or bottle of drink at room temperature can be cooled in a cooler containing ice. Adding salt and alcohol to the ice can increase the rate of cooling and affect the drink’s final temperature. Investigate how the relevant parameters affect the lowest achievable temperature and the maximum cooling rate.

Interactive Simulation (Python)

This simulation runs directly in your browser — no Python installation needed. Initial loading may take a few seconds.

دمای اولیه T0 (°C) 80 °C

دمای محیط T∞ (°C) 5 °C

τ در هوا (min) 40 min

τ در حمام یخ (min) 6 min

Press the button below to start.

Python Code Lab

Edit this starter code, enter your own experimental data, and run it directly in the browser — no Python installation required.

Python — Editable

# آزمایش «نوشیدنی سرد» — فیت τ برای چند روش سرد کردن
import numpy as np
import matplotlib.pyplot as plt

# --- بخش ۱: داده‌های شما ---
# زمان اندازه‌گیری (دقیقه)
t_min = np.array([0, 2, 5, 10, 15, 20, 30, 45])
# دما در محیط‌های مختلف (°C)
T_air  = np.array([80, 76, 70, 62, 55, 49, 41, 32])
T_ref  = np.array([80, 74, 66, 55, 46, 39, 28, 18])
T_ice  = np.array([80, 55, 32, 15,  9,  7,  6,  5])

Tinf_air = 22.0
Tinf_ref = 5.0
Tinf_ice = 0.5

# --- بخش ۲: نظریه: فیت T(t) = T∞ + (T0−T∞) exp(−t/τ) روی هر ست ---
def fit_tau(t, T, Tinf, T0):
    y = (T[T > Tinf+0.1] - Tinf)/(T0 - Tinf)
    tt = t[T > Tinf+0.1]
    ln_y = np.log(y)
    slope, _ = np.polyfit(tt, ln_y, 1)
    return -1/slope   # minutes

tau_air = fit_tau(t_min, T_air, Tinf_air, 80)
tau_ref = fit_tau(t_min, T_ref, Tinf_ref, 80)
tau_ice = fit_tau(t_min, T_ice, Tinf_ice, 80)

t_sm = np.linspace(0, 45, 200)
T_air_th = Tinf_air + (80 - Tinf_air)*np.exp(-t_sm/tau_air)
T_ref_th = Tinf_ref + (80 - Tinf_ref)*np.exp(-t_sm/tau_ref)
T_ice_th = Tinf_ice + (80 - Tinf_ice)*np.exp(-t_sm/tau_ice)

# --- بخش ۳: نمودار ---
fig, ax = plt.subplots(figsize=(8, 4.8))
ax.plot(t_min, T_air, 'ro', ms=8);  ax.plot(t_sm, T_air_th, 'r-', label=f'air τ={tau_air:.1f} min')
ax.plot(t_min, T_ref, 'gs', ms=8);  ax.plot(t_sm, T_ref_th, 'g-', label=f'fridge τ={tau_ref:.1f} min')
ax.plot(t_min, T_ice, 'b^', ms=8);  ax.plot(t_sm, T_ice_th, 'b-', label=f'ice τ={tau_ice:.1f} min')
ax.set_xlabel('time (min)'); ax.set_ylabel('drink T (°C)')
ax.set_title('Cold drink cooling curves')
ax.legend(); ax.grid(True, alpha=0.3)
plt.tight_layout()

# --- بخش ۴: نتایج ---
print(f"τ (still air)    = {tau_air:.2f} min")
print(f"τ (refrigerator) = {tau_ref:.2f} min")
print(f"τ (ice bath)     = {tau_ice:.2f} min")
print(f"nSpeed-up ice / air     = {tau_air/tau_ice:.1f}×")
print(f"Speed-up ice / fridge  = {tau_ref/tau_ice:.1f}×")

Ready to run

Output

Problem Description & Analysis

Interactive Simulation (Python)

Python Code Lab

Suggested Resources